MWR for the first- and third-order differential equations

I am teaching an introductory course on FEA this semester. Teaching always involves learning—at least on the teacher’s side.

No, there was no typo in there. I did mean what I just said. It’s based on my own personal observation. Teaching actually involves (real) learning on the part of the teacher—and hopefully, if he is effective enough in his teaching (and if the student, too, is attentive and hard-working enough), then, also on the part of the student.

When you teach a course, in thinking about how to simplify the ideas involved, how to present them better, you have to mentally go over the topics again and again; you have to think and re-think about the material; you have to see if rearranging the ideas and the concepts involved or seeing them in a different light might make it any easier to “get” it or even just to retain it, and so on. … The end result is that you often actually end up deriving at least new mnemonics if not establishing new connections about the topics. In any case, you derive better conceptual integrations or strengthen them better. You end up mastering the material better than at the beginning of the course. … Or at least that’s what happens to me. I always end up learning at least a bit more about what I am teaching.

And, sometimes, the teacher even ends up deriving completely new ideas this way. At least, it seems, I just did—about the nature of FEA and computational mechanics in general. The idea is new, at least to me. But anyway, talking about this new idea is for some other day. … I have to first rigorously think about it. The idea, as of today, is just at that nascent phase (it struck me right this evening). I plan to put it to the paper soon, work out its details, refine the idea, and put it in a more rigorous form, etc. That will take time. And then, second, I have to also check whether someone has already published something of that kind or not. … As someone—was it Mark Twain?—said, the best of my ideas were stolen by the ancients… So, that part—checking the literature—too, will take quite some time. My own anticipation is that someone must have written something about it. In any case, it’s not all that big an idea. Just a simple something.

But, anyway, in the meanwhile, for this blog post, let me note down something different. An item, not of my knowledge, and not one of even potentially new knowledge, but of my ignorance, which got highlighted recently, during my lecture preparations.

I realized that if one of my students poses a question about it, I don’t know the reason why MWR (the method of weighted residuals) isn’t effective, or at least isn’t often used, and may be even cannot be relied on, for the first- and the third-order differential equations.  (See, see, see, I don’t even know whether it’s a “cannot” or an “isn’t”!) I don’t know the answer to that question.

Of course, as it so happens, most differential equations of engineering importance are only of the second and the fourth order. Whether linear or non-linear, they simply aren’t of the third-order. I haven’t myself seen a single third-order differential equation in any of the course-work I have ever done so far. Sure, I have seen such equations, but only in a mathematical handbook on the differential equations—never in a text-book or a monograph on engineering sciences as such. And, even if of the first-order, in physics and engineering, they often come as coupled equations, and thus, (almost nonchalantly, right in front of your eyes) jump into the usual class of the second-order differential equations—e.g. the partial differential wave equation.

Anyway, coming back to this MWR-related issue, I checked up the text-books by Reddy and Finlayson, but didn’t find the reason mentioned. I hope that someone knows the answer—someone would. So, I am going to raise this issue at iMechanica, right today.

That’s about all for this blog post, folks. Once I post my question at iMechanica, may be I will come back and add a link to it from here, but that’s about it. More, some other time.

[And, yes, I promise to blog about the new idea once I am done working it out and checking about it a bit. It just struck me just today, and it still is purely in the conceptual terms. The idea itself is such that it can (very) easily be translated into proper mathematical terms, but the point is: that’s something I haven’t done yet. Let me do that over, say the next few weeks/months, and then, sure, I will come back and blog about it a bit. I mean, I will sure blog about it way, way before sending any paper to any journal or so. That’s a promise. So, bye for now…]

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